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Why Fixed-Income Claim PDEs Include Cash-Flow Terms

Article Quant Q&A · Author: A.Oreo
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Summary

The document compares pricing equations for a zero-coupon bond, a fixed-rate swap, a caplet, and a floorlet under a one-factor short-rate model. The bond equation is homogeneous, while the other three equations contain an additional term representing cash flows paid during the contract. The response explains the swap case through the Feynman–Kac framework: receiving the rate difference between the floating rate and the fixed rate generates a running payoff, which appears as a source term in the pricing equation.

This illustrates how a claim’s contractual cash flows affect its PDE even when the underlying rate dynamics are already specified. The post lists source terms for the caplet and floorlet but does not explain their derivation in detail. It also gives no model specification, boundary conditions, or worked valuation, so it is a conceptual explanation rather than a complete pricing treatment.

Key ideas

  • A zero-coupon bond has no interim contractual payments, so its pricing PDE has no source term.
  • A swap’s running payment contributes an inhomogeneous term to its valuation equation.
  • The Feynman–Kac representation connects running cash flows to source terms in a pricing PDE.
  • Caplet and floorlet source terms reflect their rate-dependent payments, though the post does not derive them.

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Fixed incomeDerivatives pricingStatistics
Full text
# Why there is some inhomogeneous term in the PDE of fixed income


# Why there is some inhomogeneous term in the PDE of fixed income












We consider one factor driving model of fixed income product say short-term interest $r(t)=\lim\limits_{T\rightarrow t} R(t,T),$ $R(t,T)$ is yield i.e $$B(t,T)e^{(T-t)R(t,T)} = 1$$ Then we see several PDE of contingent claim

`Zero-coupon bond` $B(t,T)$ $$\dfrac{\partial B}{\partial t} + LB -r(t)B = 0$$ here $L$ is the differential operator in Feynman-Kac equation.

`Swap of fixed rate` $r^*$ $$\dfrac{\partial V}{\partial t} + LV -r(t)V + (r - r^*) = 0$$

`Caplet at rate` $r*$ $$\dfrac{\partial V}{\partial t} + LV -r(t)V + \min(r,r^*) = 0$$

`Floorlet at rate` $r*$ $$\dfrac{\partial V}{\partial t} + LV -r(t)V + \max(r,r^*) = 0$$

Here $r = r(t)$ and $V(t,T,r(t))$ is the value of contingent claim which is the function of $t$ and $r$ e.g, for zero-coupon bond $V=B.$

I couldn't understand when the dynamic of $r(t)$ is given, why there are some inhomogeneous terms in the Black-Scholes equation? Can some one explain one of later three?

## Answer by user26484 (score 2)

https://quant.stackexchange.com/a/32323

Check out https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula, it is the $f(X_r,r)$ in the formula. For the swap you receive $(r-r^*) dt$ (assume notional of 1) which translates into your $f$ (inhomogeneous term).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.